How do you simplify #(6t^3 + 5t^2 + 9) / (2t + 3)#?

Question

Eva Abramson · Accepted Answer

#3t^2 - 2t^2 + 3 #

#(6t^3 + 5t^2 + 9) / (2t+3)#

#(2t+3(3t^2-2t+3)) / (2t+3)#

ANSWER: #3t^2 - 2t^2 + 3 #

You have to factorise the top that #2t + 3# directly factorises #6t^3 + 5t^2 + 9# so that it cancels the denominator giving you a value of #3t^2 - 2t^2 + 3#.

The are many ways of factorising #6t^3 + 5t^2 + 9#, polynomial division, rational root theorem.

I would recommend polynomial division as its pretty straight forward to learn and rely on. Here is a link that will teach you all about it and you can practise with: https://tutor.hix.ai

Abigail Allen · Answer

#3t^2-2t+3# #"one way is to use the divisor as a factor in the numerator"# #"consider the numerator"# #color(red)(3t^2)(2t+3)color(magenta)(-9t^2)+5t^2+9# #=color(red)(3t^2)(2t+3)color(red)(-2t)(2t+3)color(magenta)(+6t)+9# #=color(red)(3t^2)(2t+3)color(red)(-2t)(2t+3)color(red)(+3)(2t+3)cancel(color(magenta)(-9))cancel(+9)# #=color(red)(3t^2)(2t+3)color(red)(-2t)(2t+3)color(red)(+3)(2t+3)+0# #rArr(cancel((2t+3))(3t^2-2t+3))/cancel((2t+3))=3t^2-2t+3#

Jace Anderson · Answer

undefined To simplify the expression (6t^3 + 5t^2 + 9) / (2t + 3), you can use polynomial long division or synthetic division. Here is the simplified form:

3t^2 - 4t + 6 + (3)/(2t + 3)